Here we list the symbols and abbreviations used in the book, giving a short
description of their meaning and, whenever necessary, the number of the
page where they are introduced.

Ôł╝ The function on the left has the power series on the right
=
as its asymptotic expansion (p. 65)
Ôł╝s The function on the left has the power series on the right
=
as its Gevrey expansion of order s (p. 70)

Ôë║ We write m Ôë║ n whenever, after a rotation of C d making
the cuts point upward, the nth cut is to the right of the
mth one (p. 148)
Ôł×(¤„ )
An integral from a to in´¬ünity along the ray arg(u Ôł’ a) = ¤„
a
(pp. 78, 79)

duk Short for k ukÔł’1 du (p. 78)

╬┤ Short for z(d/dz) (p. 24)

╬│k (¤„ ) The path of integration following the negatively oriented
boundary of a sector of ´¬ünite radius, opening larger than
¤Ç/k and bisecting direction ¤„ (p. 80)

(╬±)n PochhammerÔÇ™s symbol (p. 22)
296 Symbols

Ak,k EcalleÔÇ™s acceleration operator (p. 176)
╦ť

╦ć╦ť
Ak,k EcalleÔÇ™s formal acceleration operator (p. 176)

A(k) (S, E ) The space of all E -valued functions that are holomorphic,
bounded at the origin and of exponential growth at most
k in a sector S of in´¬ünite radius (p. 62)

A(G, E ) The space of all E -valued functions that are holomorphic
in a sectorial region G and have an asymptotic expansion
at the origin (p. 67)

As (G, E ) The space of all E -valued functions that are holomorphic
in a sectorial region G and have an asymptotic expansion
of Gevrey order s (p. 71)

As,m (G, E ) The space of all E -valued functions that are meromorphic
in a sectorial region G and have a Laurent series as asymp-
totic expansion of Gevrey order s (p. 73)

Degree or valuation of a matrix power series in z Ôł’1 (p. 40)
╦ć
deg T (z)

E,F Banach spaces, resp. Banach algebras (p. 219)
Symbols 297

EÔł— The set of continuous linear maps from E into C (p. 219)
E [[z]] The space of formal power series whose coe´¬âcients are
in E (p. 64)
E [[z]]s The space of formal power series with coe´¬âcients in E and
Gevrey order s (p. 64)
E {z} The space of convergent power series whose coe´¬âcients are
in E (p. 64)
E {z}k,d The space of power series with coe´¬âcients in E that are
k-summable in direction d (p. 102)
E {z}k The space of power series with coe´¬âcients in E that are
k-summable in all but ´¬ünitely many directions (p. 105)
E {z}T,d The space of power series with coe´¬âcients in E that are
T -summable in direction d (p. 108)
E {z}T ,d The space of power series with coe´¬âcients in E that are
T -summable in the multidirection d (p. 161)
e EulerÔÇ™s constant (= exp[1])
e, ╦ć
╦ć0 The formal power series whose constant term is e, resp. 0,
while the other coe´¬âcients are equal to 0 (pp. 64, 70)
e1 Ôł— e2 The convolution of kernel functions (p. 160)
E╬± (z) Mittag-Le´¬„erÔÇ™s function (p. 233)
F (╬±; ╬▓; z) Con´¬‚uent hypergeometric function (p. 22)
F (╬±, ╬▓; ╬│; z) Hypergeometric function (p. 26)
Similar notation is used for the generalized con´¬‚uent hy-
pergeometric function (p. 23)
resp. generalized hypergeometric function (p. 26)
resp. generalized hypergeometric series (p. 107)
FFS Short for formal fundamental solution (p. 131)
f Ôł—k g Convolution of functions f and g (p. 178)
╦ć╦ć ╦ć
f Ôł—k g Convolution of formal power series f and g
╦ć (p. 178)
G A region in the complex domain, resp, on the Riemann
surface of the logarithm (p. 2)
G(d, ╬±) A sectorial region with bisecting direction d and opening
╬± (p. 61)
298 Symbols

H(G, E ) The space of functions, holomorphic in G, with values in
E (p. 221)

HLFFS Short for highest-level formal fundamental solution (p. 55)

HLNS Short for highest-level normal solution (p. 138)

J The linear map that maps functions to their asymptotic
expansion (p. 67)

J┬µ (z) BesselÔÇ™s function (p. 23)

j0 Number of singular directions in a half-open interval of
length 2¤Ç (p. 137)

j1 Number of singular directions in a half-open interval of
length ┬µ¤Ç/(qr Ôł’ p) (p. 137)

Lk The Laplace operator of order k (p. 78)
╦ć
Lk The formal Laplace operator of order k (p. 79)